Questions tagged [stochastic-differential-equations]
Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).
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how to do such stochastic integration $dS = a S^b dt + c S dW$?
How to do stochastic integration $dS = a S^b dt + c S dW$, where $a$, $b$ and $c$ are constant, $b > 0$, and $W$ is the Wiener process.
I know how to do integration for $dS = aS dt + cS dW$, or $...
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Solution to General Linear SDE
In order to find a solution for the general linear SDE
\begin{align}
dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t,
\end{align}
I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
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5
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3
answers
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Infinitesimal generator of the Brownian motion on a sphere
As explained here, the infinitesimal generator of a 1D Brownian motion is $\frac{1}{2}\Delta$. As discussed here, for the Brownian motion on circle we can write
$$Y_1=\cos(B) \\
Y_2= \sin(B)$$
and ...
- 107
4
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2
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Using Markov Property in solving PDE/SDE
I am solving the (boundary?) value problem (from Bjork I think, see below)
By Feynman-Kac, any solution has the form of a conditional expectation
$$F(t,x) = E[\psi(X_T)|X_t = x]$$
where
$$\psi(x) =...
- 12.7k
3
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1
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Properties of the time integral of Ito process
Consider the Ito process $X_t$ defined by
$$
dX_t = a(t,X_t) dt + b(t,X_t) dW_t
$$
where $W_t$ is the standard continuous-time Wiener process. Let's define the process $Y_t$ to be some integral of $...
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1
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Difference between weak ( or martingale ) and strong solutions to SDEs
Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE :
$$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$
...
- 1,980
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2
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Name of the formula transforming general SDE to linear
For SDE's of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar ...
- 4,751
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2
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Solving the SDE $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$
How to solve
$dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
- 553
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Convergence of the distribution of the Langevin diffusion to its invariant measure
Let $(X_t)_{t\ge0}$ be a solution of $${\rm d}X_t=-h'(X_t){\rm d}t+\sqrt 2W_t,\tag1$$ where $(W_t)_{t\ge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. Assume ...
- 13.5k
2
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Is the transition semigroup of the solution of an SDE with Lipschitz coefficients strongly continuous on $C_b$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous (and hence at most of linear growth) and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\...
- 13.5k
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1
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Solve the linear SDE $dX_t = aX_t \, dt +(b+cX_t) \, dW_t$
I am trying to find the solution to the SDE:
$$
dX_t=aX_tdt+(b+cX_t)dW_t
$$
for $t\ge0$, $X_0>0$, constants $a,b,c$
Would appreciate any hints as to how to approach this using ito's formula, I'm ...
- 5,654
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How is the simulation done of the black scholes model?
This post contains additional questions from:
How do I implement the Euler scheme for this SDE?
I think it is more appropriate to start a new post.
In an assignment the following SDE needs to be ...
- 598
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1
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4th order Runge-Kutta Scheme for Stochastic Differential Equations
In the book "Introduction to Stochastic Differential Equations" by Thomas C. Gard, they talk about higher-order runge-kutta type schemes. The SDE in question is a general Ito SDE of the form:
$$dX = ...
- 3,566
14
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5
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Why do people write stochastic differential equations in differential form?
I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
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0
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proving equalities in stochastic calculus
I am struggling with this question:
FIRST PART (almost done, but stuck somewhere):
Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by
the formula
\begin{equation}
...
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6
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1
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Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$
Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$
Provided Question
The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
- 576
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Possible Close-form solution for 2 dimensional $\frac{d\Sigma}{dt} = \mathbf{A}\Sigma + \Sigma \mathbf{A}^T + \mathbf{B}\mathbf{B}^T$
When $\mathbf{A},\mathbf{B}$ are 2x2 matrix and $\Sigma(t)$ are PSD, can we expect a close form solution for the following ODE,
$$
\frac{d\Sigma}{dt} = \mathbf{A}\Sigma + \Sigma \mathbf{A}^T + \mathbf{...
- 255
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Solving this SDE $dX_t = aX_tdt + bdW_t$, $X_0 = x$ to find $E[X_t^2]$
slightly related:Is there a specific term for this SDE?
Using $f(x_t, t ) = x_te^{at}$, then $df(x_t,t) = ax_te^{at} + e^{at}dx_t$
We can plug in the original question, $dX_t = aX_tdt + bdW_t$ which ...
4
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1
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Do you know this result about Conditional Expectation?
Let $X,Y$ real independent random variables and $g: \mathbb R^2\to \mathbb R$ an integrable or positive function. then
$$\mathbb E[g(X,Y)|Y]=\phi(Y)$$
where $\phi(y):=\mathbb E[g(X,y)]$.
This is ...
- 1,139
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votes
1
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Geometric Brownian motion, product ansatz rationale
My question is why does the subsequent product ansatz for the geometric Brownian motion work?
Suppose we have the gBm
$$dS_t=\mu S_tdt+\sigma S_tdB_t,\ S(0)=S_0$$
We assume the solution is given by ...
- 37
3
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2
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What's the solution of Stock price based on GBM model?
Stock price has a classic model based on GBM:
$$dS = \mu S dt + \sigma S dW$$
based on this call options values could be solve -- Black-Scholes formula.
But, what is the solution for the Stock ...
- 4,751
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multivariate Ito isometry
I wonder whether there exists a straightforward extension of the Ito isometry to multidimensional processes.
In the one-dimensional case the Ito isometry can be written as
$\mathbb{E}[ (\int_0^T X_t ...
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Integral of "white noise" multiplied by exponential term (multivariate Ornstein-Uhlenbeck)
Consider a matrix differential equation for the vector $\mathbf{y}$ of the form
\begin{equation}
\dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, ,
\end{equation}
where $A$ is a ...
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Using Itos Lemma To Derive an Ito Stochastic Differential Equation
So we have an Ito Stochastic Differential Equation with $b$ as a constant: $$dX_t = (bX_t +1)dt +2 \sqrt{X_t}dW_t $$
I then am told to let $Y_t = \sqrt{X_t} $ and thus derive the Ito stochastic ...
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Determining Properties of Stochastic Differential Equation as Drift Tends to $0$
Consider the stochastic differential equation
$$dS=\left(\mu S+\frac{\Lambda-S}{\omega}\right)dt+\sigma S dW_t.$$
such that $W_t$ is a Wiener process and $\mu,\sigma,\Lambda,\omega\in\mathbb{R}^+$. As ...
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2
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Find the covariance of $x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$
I want to find the covariance of the following SDE:
$$x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$$
To start, I find the mean, it is simply:
$$ E[x_{t}] = E[x_{0}] e^{-\...
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1
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Find closed-form solution to the SDE $dX_t = dt + 2 \sqrt{X_t} \, dW_t$
I've been tasked to solve the following SDE:
$dX_t = dt + 2\sqrt{X_t}dW_t, \ \ \ t \in \mathbb{R}_+$
where $W_t$ is a standard Brownian motion and $X_0=x$.
I need a closed form solution.
What is ...
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1
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Finding $X_t$ of an Itô Diffusion [closed]
Someone please help me with this:
I have that $X_t$ is the Ito diffusion with genertator $A(f)(x)=\alpha xf'(x)+f''(x).$
Then, if $X_0=x \in \mathbb{R}^+$, how do I find $X_t$?
- 23
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0
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Is the strong solution of a SDE adapted to the filtration generated by the driving Brownian motion?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$\xi$ be an $\...
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1
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Rewriting sum of correlated Brownian Motions as a single brownian motion
Say I have some stochastic differential equation:
$$dX_t = \alpha dt + \sigma_1 dW^1_t + \sigma_2dW^2_t$$
where $W^1_t$ and $W^2_t$ are standard Brownian Motions where $cov(dW^1_t, dW^2_t) = \rho dt$....
- 3,256
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Solution to the linear SDE $dX_t = \alpha X_t \, dt + \sqrt{2} dB_t$ using Itô calculus
So if I have the following generator and an initial condition:
$$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$
I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
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Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]
Solve the following stochastic differential equations
$ dX_t = \frac{1}{2 X_t} dt + dB_t$
or equivalently with a transformation $Y_t = X_t^2$
$ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
- 1,302
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Initial Distribution of Stochastic Differential Equations
consider the SDE
\begin{align}
\begin{cases}
X_t= \mu (t,x_t)dt + \sigma(t,X_t) d W_t \quad \forall t\in [0,T] \ (\text{or } t\geq 0),\\
X_0 \sim \xi.
\end{cases}
\end{align}
Suppose that, ...
- 380
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votes
1
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Is an SDE really equal to an integral equation, or is it rather "its integral" that is?
Ive been told and been reading in some textbooks on SDE's that an SDE really is an integral equation. In other words, that
$ dX= \beta dt + \sigma dW$ $\,$ "really means" $\,$ $X_{t}= X_{0} +\int_{0}...
- 1,755
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Stochastic Differential Equation solution for Geometric Brownian Motion
I am having difficulty in understanding the solution for a GBM given the SDE:
$$dY(t)=\mu \ Y(t) \ dt + \sigma \ Y(t) \ dZ(t)$$ or $$\frac{dY(t)}{Y(t)}=\mu \ dt + \sigma \ dZ(t)$$
The solution for ...
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1
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Discretizing a Stochastic Volatility SDE
How does the discrete time stochastic volatility model arise from the continuous time one?
I have the following continuous time stochastic volatility model. $S_t$ is the price, and $v_t$ is a ...
- 533
8
votes
2
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Justifying "multiplying by a differential" or a random variable in a stochastic differential equation
This is a very basic question relating to why we are allowed to multiply by random variables within an SDE. Every text/notes set that I've seen does the following, for $X_t$ continuous + adapted and $...
- 1,408
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0
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Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.
Let$^1$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$U$ be a separable Hilbert space
$Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace
$(W_t)_{t\ge0}...
- 13.5k
7
votes
1
answer
387
views
Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations?
I am reading some introductory material on stochastic differential equations at the moment. In almost all cases, the equations which are presented are of the form
$$ dX_t = \mu(t,X_t) dt + \sigma(t, ...
- 1,694
7
votes
2
answers
714
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Definition of Stratonovich Integral
I have a doubt in definition of the Stratonovich integral. In "Stochastic Calculus for Finance" by Steven Shreve, he defines it using the midpoint $\frac {(t_i+t_{i+1})}{2}$ of the ...
- 81
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votes
1
answer
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Infinitesimal Generator for Stochastic Processes
Suppose one has the an Ito process of the form:
$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$
The infinitesimal generator $LV(x)$ is defined by:
$$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) \right]-V(x)}{...
- 3,566
6
votes
2
answers
393
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Expected value of $S_t$ where $dS_t=(\mu S_t+a)dt + (\sigma S_t +b)dW_t$
I have the stochastic differential equation: $$dS_t=(\mu S_t+a)dt + (\sigma S_t +b)dW_t$$
where $W_t$ is a Wiener process with $S_0 > 0$ and $\mu, \sigma, a, b \in \mathbb{R}$.
I have found the ...
- 99
5
votes
1
answer
2k
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Solution of SDE $dX_t = \mu(t)X_tdt + \sigma X_t dW_t$
I didn't study stochastic process in a systematic way, but I need to use it in financial analysis. Here's my question.
I know the solution of the SDE $dX_t = \mu X_tdt + \sigma X_t dW_t$, given that $...
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5
votes
0
answers
189
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Are SDE's really "differential"?
An SDE of the form
$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$
is really short-hand notation for an equation involving Ito integrals:
$$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,...
- 12.2k
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0
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How to solve system of stochastic differential equations?
I have the following two SDEs
$$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$
$$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$
where $W$ is the standard Brownian motion/Wiener process. This isn't homework, I'm ...
- 123
5
votes
1
answer
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Expectation of stochastic differential equation
I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
- 53
4
votes
1
answer
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Do there exist diffusions that do not solve any SDE?
Diffusions are continuous time stochastic processess having continuous paths and satisfying the strong Markov property.
I know it is possible to characterize some diffusion processes as solutions to ...
- 364
4
votes
3
answers
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If $g(x,y)$ measurable, why $g$ can be boundedly approximate by functions of the form $\sum_{k=1}^n f_k(x)h_k(y)$?
Let $g=g(x,y)$ measurable.
1) What does mean "$g$ can be boundedly approximate by the sequence $g_n$" ? What is this "boundedly" ?
2) Why $g$ can be boundedly approximate by functions of the form $\...
- 980
4
votes
1
answer
875
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Strong solutions SDE inequality with an application of Gronwall's inequality
Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by:
$$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$
where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
user155670
4
votes
1
answer
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Stochastic Differential Equations - connection between white noise and Wiener process
I am trying to understand the connection between white noise and the Wiener process in the context of SDEs. At the beginning one starts with a differential equation including white noise $\xi_t$, e.g.,...
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